Solving larger Travelling Salesman Problem networks with a penalty-free Variational Quantum Algorithm

• URL: http://arxiv.org/abs/2512.06523v1
• Date: Sat, 06 Dec 2025 18:21:21 GMT
• Title: Solving larger Travelling Salesman Problem networks with a penalty-free Variational Quantum Algorithm
• Authors: Daniel Goldsmith, Xing Liang, Dimitrios Makris, Hongwei Wu,
• Abstract summary: The Travelling Salesman Problem (TSP) is a well-known NP-Hard problem, with industrial use cases such as last-mile delivery.<n>We present high quality solutions in noise-free simulations of networks with up to twelve locations using a hybrid penalty-free, circuit-model.
• Score: 5.690622599243828
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Travelling Salesman Problem (TSP) is a well-known NP-Hard combinatorial optimisation problem, with industrial use cases such as last-mile delivery. Although TSP has been studied extensively on quantum computers, it is rare to find quantum solutions of TSP network with more than a dozen locations. In this paper, we present high quality solutions in noise-free Qiskit simulations of networks with up to twelve locations using a hybrid penalty-free, circuit-model, Variational Quantum Algorithm (VQA). Noisy qubits are also simulated. To our knowledge, this is the first successful VQA simulation of a twelve-location TSP on circuit-model devices. Multiple encoding strategies, including factorial, non-factorial, and Gray encoding are evaluated. Our formulation scales as $\mathcal{O}(nlog_2(n))$ qubits, requiring only 29 qubits for twelve locations, compared with over 100 qubits for conventional approaches scaling as $\mathcal{O}(n^2)$. Computational time is further reduced by almost two orders of magnitude through the use of Simultaneous Perturbation Stochastic Approximation (SPSA) gradient estimation and cost-function caching. We also introduce a novel machine-learning model, and benchmark both quantum and classical approaches against a Monte Carlo baseline. The VQA outperforms the classical machine-learning approach, and performs similarly to Monte Carlo for the small networks simulated. Additionally, the results indicate a trend toward improved performance with problem size, outlining a pathway to solving larger TSP instances on quantum devices.

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